Related papers: Intercenter Geometry
We present a new model of a non-Euclidean plane, in which angles in a triangle sum up to $\pi$. It is a subspace of the Cartesian plane over the field of hyperreal numbers $\mathbb{R}^*$. The model enables one to represent the negation of…
Author developed a uniform model for different spaces where distance and angle measure kinds are parameters. This model is calculus centric, but can also be used in theoretical research. It is useful in the following domains: deduction of…
Four points ordered in the positive order on the unit circle determine the vertices of a quadrilateral, which is considered either as a euclidean or as a hyperbolic quadrilateral depending on whether the lines connecting the vertices are…
We propose a novel theoretical framework for barycentric interpolation, using concepts recently developed in mathematical physics. Generalized barycentric coordinates are defined similarly to Shepard's method, using positive geometries -…
By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are…
We show that Euclidean geometry in suitably high dimension can be expressed as a theory of orthogonality of subspaces with fixed dimensions and fixed dimension of their meet.
An isometry is a geometric transformation that preserves distances between pairs of points. We present methods to classify isometries in the Euclidean plane, and extend these methods to spherical, single elliptical, and hyperbolic geometry.…
The purpose of this paper is to present projective geometry in a synthetic, visual and intuitive style through the central notion of harmonicity which leads to harmonic curves. This presentation includes new results, unpublished proofs of…
One of the best things about geometry is that it's cool! Geometry enables us to create incredible designs and astounding patterns. This article shows how to use a simple technique (iteration) to create designs that are both cool and…
In this article, I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us (the…
A new method of metric space investigation, based on classification of its finite subspaces, is suggested. It admits to derive information on metric space properties which is encoded in metric. The method describes geometry in terms of only…
This paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra…
The concept of number and its generalization has played a central role in the development of mathematics over many centuries and many civilizations. Noteworthy milestones in this long and arduous process were the developments of the real…
Coarse geometry studies metric spaces on the large scale. Our goal here is to study dynamics from a coarse point of view. To this end we introduce a coarse version of topological entropy, suitable for unbounded metric spaces, consistent…
In the era of foundation models and Large Language Models (LLMs), Euclidean space has been the de facto geometric setting for machine learning architectures. However, recent literature has demonstrated that this choice comes with…
Geometric torsions are torsions of acyclic complexes of vector spaces which consist of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a…
Many problems in Euclidean geometry, arising in computational design and fabrication, amount to a system of constraints, which is challenging to solve. We suggest a new general approach to the solution, which is to start with analogous…
In this paper, we propose to study a new geometric optimization problem called "geometric prototype" in Euclidean space. Given a set of patterns, where each pattern is represented by a (weighted or unweighted) point set, the geometric…
Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…
This work presents a new cyclic architecture that extracts high-frequency patterns from images and re-insert them as geometric features. This procedure allows us to enhance the resolution of low-cost depth sensors capturing fine details on…